Applying Theory to Practice

Each of the theoretical frameworks outlined above offers a useful perspective upon various aspects of the process of problem posing and problem solving. The SOLO taxonomy provides an effective but simple classification system by which student responses may be distinguished into various levels of complexity and understanding - a very useful tool when attempting to assess problem solving. The van Hiele model potentially offers a framework for teachers to assist their students to move to higher cognitive ground, and the theories based upon the work of Vygotsky are helpful in drawing attention to the significance of scaffolding and support in the learning process. Each explicitly values the role of language in learning and understanding - students must be able to put concepts into their own words if their teachers are to be sure they have not simply learned to "mimic" the language they have heard without internalising its meaning. Most importantly, each emphasises the vital role of challenge in the learning process. The learner must be engaged by the task: interested and motivated to pursue a solution (indeed, ideally, learners should become accustomed to considering multiple solutions, and to verify these using multiple strategies).

Taking these issues into account suggests an additional component to the problem posing and problem solving process - that of designing a supportive structure to complement the various problems offered to students. It is suggested here that this might take two forms associated with each problem situation offered: a "graduated help" facility, by which students may select from increasing levels of help for the task and, complementing this, a "structured solution" format, in which guidance is offered in constructing an appropriate solution. Both confront two significant sources of uncertainty for many learners - at the first stage, when the problem or task is first encountered (knowing just what to do to begin) and, then, the blank page upon which an answer or solution must be created. This latter approach is addressed first.

The SOLO Taxonomy. A structured solution format.